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An ellipsoid is a three-dimensional geometric shape that resembles a stretched or compressed sphere. The equation used to depict an ellipsoid in a coordinate system is quite similar to that of a sphere but includes different denominators for each variable to represent its unique dimensions. The standard form of an ellipsoid equation is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \] This equation is quite versatile as it describes how the ellipsoid stretches along the x, y, and z axes. It's important to note:

• The center of the ellipsoid is at the origin (0, 0, 0).
• The semi-axes lengths are \(a\), \(b\), and \(c\), respectively.
• This form helps in visualizing the spread of the ellipsoid in different directions.

In the given problem, the equation takes the form \(x^{2}+(y/2)^{2}+(z/c)^{2}=1\). This implies the semi-axes lengths are 1, 2, and \(c\) for the x, y, and z dimensions, respectively. The form matches the standard ellipsoid equation, indicating an elongation along specific axes, determined by the denominators.

Calculating the volume of an ellipsoid is a fascinating extension of the formula for a sphere's volume. Unlike a sphere that has the same radius in all directions, an ellipsoid has different semi-axis lengths along each coordinate direction. Therefore, its volume can be computed using the following formula:\[ V = \frac{4}{3} \pi a b c \]Here's a quick explanation of each component in the formula:

• \(\frac{4}{3}\pi\): This fraction is derived from the volume formula of a sphere, accounting for the three-dimensional spread of the shape.
• \(a\), \(b\), and \(c\): These variables represent the semi-axes lengths of the ellipsoid in the x, y, and z directions.

For our exercise, the semi-axes are \(a = 1\), \(b = 2\), and \(c = c\). Substituting these values into the volume formula results in:\[ V = \frac{4}{3} \pi (1) (2) (c) = \frac{8}{3} \pi c \]This tells us how the size of the ellipsoid changes with its axis dimensions. Setting this expression equal to the required volume simplifies finding the missing value for \(c\).

The semi-axes of an ellipsoid are fundamental to understanding its overall shape and orientation. These semi-axes are, essentially, half the lengths of the ellipsoid along each principal axis. In simpler terms, they are the radii of the ellipsoid in the x, y, and z directions:

• Semi-major axis: The longest axis of the ellipsoid.
• Semi-minor axis: The shortest axis of the ellipsoid.
• Intermediate axis: An axis that is neither the longest nor the shortest.

In the context of our given ellipsoid equation, \(x^{2}+(y/2)^{2}+(z/c)^{2}=1\), the semi-axes are specified as:

• \(a = 1\)
• \(b = 2\)
• \(c\), which needs to be determined based on the volume requirement.

These semi-axes tell us that the ellipsoid stretches further in the y direction than in the x or z directions, assuming \(c\) is not larger than 2. Understanding these axial lengths is key to visualizing the orientation and the extent of an ellipsoid in three-dimensional space.